Someone I know is trying to figure out if the following concepts already have an established name in the literature, and MO is a great place to ask around.

1) Suppose $X$ is a metric space equipped with an associative product
and a unit element.
Let $m: X \times X \to X$ be the product.
Suppose also that $m$ is nonexpansive, i.e.
$$
d\big( m(x,x'),m(y,y') \big) \le d\big((x,x'),(y,y')\big)
$$
say when $X\times X$ is given
the Euclidean metric with respect to the given metric of $X$.
Is there a standard name for this type of structure?

2) Let $f : X \to Y$ be a map between metric spaces with the property that for all $x,y \in X$
$$
d(f(x), f(y)) \geq d(x,y).
$$
So in a way this is the opposite of a nonexpansive map.
Is there a name for such maps?